From spherical to Euclidean illumination (Q2189422)

The unit sphere \(S^{d}\), in the \(d\)-dimensional Euclidean space, is called the \(d\)-dimensional spherical space. A subset of \(K\subset S^d\) is spherically convex if it is contained in an open hemisphere of \(S^d\) moreover, for any two points of the set the shorter unit circle arc connecting them belongs to the set. The authors introduce the following new concept of illumination. Let \(K \subset S^d\) be a convex body, and let \(p \in S^{d}\setminus K\). A boundary point \(q \in bd{K}\) is said to be illuminated from \(p\) if it is not antipodal to \(p\), the spherical segment with endpoints \(p\) and \(q\) does not intersect the interior \(int(K)\) of \(K\), and the great circle through \(p\) and \(q\) does. \(K\) is said to be illuminated from a set \(S \subset S^{d}\setminus K\), if every boundary point of \(K\) is illuminated from at least one point of \(S\). The smallest cardinality of a set \(S\) that illuminates \(K\) and lies on a \((d-1)\)-dimensional great sphere of \(S^d\) which is disjoint from \(K\), is called the illumination number of \(K\) in \(S^d\). A class of spherically convex bodies in \(S^d\), \(d>2\), such as its illumination number is \(d+1\) is identified. The combinatorial illumination number is also defined and a class of spherically convex bodies in \(S^d\), \(d>2\), such as its combinatorial illumination number is \(d+1\) is also identified. In case of \(d=3\), the combinatorial class of a spherically convex polyhedron contains a Koebe polyhedron with the combinatorial illumination number equal to 4.